Dicot family: Difference between revisions

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The [[5-limit]] parent [[comma]] for the '''dicot family''' is [[25/24]], the classical chromatic semitone. Its [[monzo]] is {{monzo| -3 -1 2 }}, and flipping that yields {{multival| 2 1 -3 }} for the [[wedgie]]. This tells us the [[generator]] is a classical third (major and minor mean the same thing), and that two such thirds give a fifth. In fact, (5/4)<sup>2</sup> = (3/2)(25/24).  
{{Technical data page}}
The '''dicot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] [[25/24]], the classical chromatic semitone.  


Possible tunings for dicot are [[7edo]], [[17edo]], [[24edo]] using the val {{val| 24 38 55 }} (24c) and [[31edo]] using the val {{val| 31 49 71 }} (31c). In a sense, what dicot is all about is using neutral thirds and pretending that is 5-limit, and like any temperament which seems to involve pretending, dicot is at the edge of what can sensibly be called a temperament at all. In other words, it is an [[exotemperament]].
== Dicot ==
The head of this family, dicot, is [[generator|generated]] by a classical third (major and minor mean the same thing), and two such thirds give a fifth. In fact, {{nowrap|(5/4)<sup>2</sup> {{=}} (3/2)(25/24)}}. Its [[ploidacot]] is the same as its name, dicot.
 
Possible tunings for dicot are [[7edo]], [[10edo]], [[17edo]], [[24edo]] using the val {{val| 24 38 55 }} (24c), and [[31edo]] using the val {{val| 31 49 71 }} (31c). In a sense, what dicot is all about is using neutral thirds and sixths and pretending that these are 5-limit, and like any temperament which seems to involve a lot of "pretending", dicot is close to the edge of what can be sensibly called a temperament at all. In other words, it is an [[exotemperament]].


== Dicot ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 12: Line 15:
: mapping generators: ~2, ~5/4
: mapping generators: ~2, ~5/4


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 348.594
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1206.283{{c}}, ~6/5 = 350.420{{c}}
: [[error map]]: {{val| +6.283 +5.167 -23.328 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 351.086{{c}}
: error map: {{val| 0.000 +0.216 -35.228 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 5-odd-limit [[diamond monotone]]: ~5/4 = [300.000, 400.000] (1\4 to 1\3)
* [[5-odd-limit]] [[diamond monotone]]: ~5/4 = [300.000, 400.000] (1\4 to 1\3)
* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [315.641, 386.314] (full comma to untempered)
* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [315.641, 386.314] (full comma to untempered)
* 5-odd-limit diamond monotone and tradeoff: ~5/4 = [315.641, 386.314]


{{Optimal ET sequence|legend=1| 3, 4, 7, 17, 24c, 31c }}
{{Optimal ET sequence|legend=1| 3, 4, 7, 17, 24c, 31c }}


[[Badness]]: 0.013028
[[Badness]] (Sintel): 0.306


=== Overview to extensions ===
=== Overview to extensions ===
The second comma of the [[Normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. Septimal dicot, with wedgie {{multival| 2 1 3 -3 -1 4 }} adds 36/35, sharp with wedgie {{multival| 2 1 6 -3 4 11 }} adds 28/27, and dichotic with wedgie {{multival| 2 1 -4 -3 -12 -12 }} adds 64/63, all retaining the same period and generator.  
The second comma of the [[normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. Septimal dicot adds [[36/35]], flattie adds [[21/20]], sharpie adds [[28/27]], and dichotic adds [[64/63]], all retaining the same period and generator.  


Decimal with wedgie {{multival| 4 2 2 -6 -8 -1 }} adds 49/48, sidi with wedgie {{multival| 4 2 9 -3 6 15 }} adds 245/243, and jamesbond with wedgie {{multival| 0 0 7 0 11 16 }} adds 81/80. Here decimal divides the period to 1/2 octave, and sidi uses 9/7 as a generator, with two of them making up the combined 5/3 and 8/5 neutral sixth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.
Decimal adds [[49/48]], sidi adds [[245/243]], and jamesbond adds [[16/15]]. Here decimal divides the [[period]] to a [[sqrt(2)|semi-octave]], and sidi uses 14/9 as a generator, with two of them making up the combined 5/2~12/5 neutral tenth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.
 
Temperaments discussed elsewhere are:
* ''[[Geryon]]'' → [[Very low accuracy temperaments #Geryon|Very low accuracy temperaments]]
* ''[[Jamesbond]]'' → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]]
 
The rest are considered below.  


=== 2.3.5.11 subgroup ===
=== 2.3.5.11 subgroup ===
The 2.3.5.11-subgroup extension maps [[11/9]]~[[27/22]] to the neutral third. As such, it is related to most of the septimal extensions.


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


[[Comma list]]: 25/24, 45/44
Comma list: 25/24, 45/44
 
[[Gencom]]: [2 5/4; 25/24 45/44]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 2 0 2}}, {{val|0 2 1 0 5}}]


[[Mapping|Sval mapping]]: [{{val|1 1 2 2}}, {{val|0 2 1 5}}]
Subgroup val mapping: {{mapping| 1 1 2 2 | 0 2 1 5 }}


[[Tp tuning|POL2 generator]]: ~5/4 = 346.734
Gencom mapping: {{mapping| 1 1 2 0 2 | 0 2 1 0 5 }}


{{Optimal ET sequence|legend=1| 3e, 4e, 7, 24c, 31c, 38cc, 45cce }}
Optimal tunings:
* WE: ~2 = 1206.750{{c}}, ~6/5 = 348.684{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 348.954{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 5.621 cents
{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }}


Related temperaments: [[Dicot family #Septimal dicot|dicot]], [[Dicot family #Sharp|sharp]], [[Dicot family #Dichotic|dichotic]]
Badness (Sintel): 0.370


==== 2.3.5.11.13 ====
==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13
Subgroup: 2.3.5.11.13


[[Comma list]]: 25/24, 40/39, 45/44
Comma list: 25/24, 40/39, 45/44


[[Gencom]]: [2 5/4; 25/24 40/39 45/44]
Subgroup val mapping: {{mapping| 1 1 2 2 4 | 0 2 1 5 -1 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 2 0 2 4}}, {{val|0 2 1 0 5 -1}}]
Gencom mapping: {{mapping| 1 1 2 0 2 4 | 0 2 1 0 5 -1 }}


[[Mapping|Sval mapping]]: [{{val|1 1 2 2 4}}, {{val|0 2 1 5 -1}}]
Optimal tunings:  
* WE: ~2 = 1202.433{{c}}, ~5/4 = 351.237{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 350.978{{c}}


[[Tp tuning|POL2 generator]]: ~5/4 = 350.526
{{Optimal ET sequence|legend=0| 3e, 7, 17 }}


{{Optimal ET sequence|legend=1| 3e, 7, 17, 24c }}
Badness (Sintel): 0.536


[[Tp tuning #T2 tuning|RMS error]]: 5.916 cents
== Septimal dicot ==
Septimal dicot is the extension where [[7/6]] and [[9/7]] are also conflated into 5/4~6/5. Although 5/4~6/5 covers a giant block of pitches already, 7/6 and 9/7 are often considered as thirds too. On that account one could argue for the canonicity of this extension, despite the relatively poor accuracy.  


== Septimal dicot ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 72: Line 85:
{{Mapping|legend=1| 1 1 2 2 | 0 2 1 3 }}
{{Mapping|legend=1| 1 1 2 2 | 0 2 1 3 }}


{{Multival|legend=1| 2 1 3 -3 -1 4 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1205.532{{c}}, ~6/5 = 337.931{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 336.381
: [[error map]]: {{val| +5.532 -20.561 -37.319 +56.032 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.561{{c}}
: error map: {{val| 0.000 -24.834 -47.753 +46.856 }}


{{Optimal ET sequence|legend=1| 3d, 4, 7, 18bc, 25bccd }}
{{Optimal ET sequence|legend=1| 3d, 4, 7 }}


[[Badness]]: 0.019935
[[Badness]] (Sintel): 0.504


=== 11-limit ===
=== 11-limit ===
Line 87: Line 102:
Mapping: {{mapping| 1 1 2 2 2 | 0 2 1 3 5 }}
Mapping: {{mapping| 1 1 2 2 2 | 0 2 1 3 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 342.125
Optimal tunings:
* WE: ~2 = 1203.346{{c}}, ~6/5 = 343.078{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 343.260{{c}}


{{Optimal ET sequence|legend=1| 3de, 4e, 7 }}
{{Optimal ET sequence|legend=0| 3de, 4e, 7 }}


Badness: 0.019854
Badness (Sintel): 0.656


=== Eudicot ===
=== Eudicot ===
Line 100: Line 117:
Mapping: {{mapping| 1 1 2 2 4 | 0 2 1 3 -2 }}
Mapping: {{mapping| 1 1 2 2 4 | 0 2 1 3 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 336.051
Optimal tunings:
* WE: ~2 = 1205.828{{c}}, ~6/5 = 337.683{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 336.909{{c}}


{{Optimal ET sequence|legend=1| 3d, 4, 7, 18bc, 25bccd }}
{{Optimal ET sequence|legend=0| 3d, 4, 7, 18bc, 25bccd }}


Badness: 0.027114
Badness (Sintel): 0.896


==== 13-limit ====
==== 13-limit ====
Line 113: Line 132:
Mapping: {{mapping| 1 1 2 2 4 4 | 0 2 1 3 -2 -1 }}
Mapping: {{mapping| 1 1 2 2 4 4 | 0 2 1 3 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 338.846
Optimal tunings:
* WE: ~2 = 1202.660{{c}}, ~6/5 = 339.597{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 339.104{{c}}


{{Optimal ET sequence|legend=1| 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef }}
{{Optimal ET sequence|legend=0| 3d, 4, 7 }}


Badness: 0.023828
Badness (Sintel): 0.985
 
== Flattie ==
This temperament used to be known as ''flat''. Unlike septimal dicot where 7/6 is added to the neutral third, here [[8/7]] is added instead.


== Flat ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 126: Line 149:
{{Mapping|legend=1| 1 1 2 3 | 0 2 1 -1 }}
{{Mapping|legend=1| 1 1 2 3 | 0 2 1 -1 }}


{{Multival|legend=1| 2 1 -1 -3 -7 -5 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1220.466{{c}}, ~6/5 = 337.577{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 331.916
: [[error map]]: {{val| +20.466 -6.335 -7.804 -45.004 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 335.391{{c}}
: error map: {{val| 0.000 -31.173 -50.922 -104.217 }}


{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}


[[Badness]]: 0.025381
[[Badness]] (Sintel): 0.642


=== 11-limit ===
=== 11-limit ===
Line 141: Line 166:
Mapping: {{mapping| 1 1 2 3 4 | 0 2 1 -1 -2 }}
Mapping: {{mapping| 1 1 2 3 4 | 0 2 1 -1 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 337.532
Optimal tunings:
* WE: ~2 = 1216.069{{c}}, ~6/5 = 342.052{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 338.467{{c}}


{{Optimal ET sequence|legend=1| 3, 4, 7d }}
{{Optimal ET sequence|legend=0| 3, 4, 7d }}


Badness: 0.024988
Badness (Sintel): 0.826


=== 13-limit ===
=== 13-limit ===
Line 154: Line 181:
Mapping: {{mapping| 1 1 2 3 4 4 | 0 2 1 -1 -2 -1 }}
Mapping: {{mapping| 1 1 2 3 4 4 | 0 2 1 -1 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 341.023
Optimal tunings:
* WE: ~2 = 1211.546{{c}}, ~6/5 = 344.304{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 341.373{{c}}


{{Optimal ET sequence|legend=1| 3, 4, 7d }}
{{Optimal ET sequence|legend=0| 3, 4, 7d }}


Badness: 0.023420
Badness (Sintel): 0.968
 
== Sharpie ==
This temperament used to be known as ''sharp''. This is where you find 7/6 at the major second and [[7/4]] at the major sixth.


== Sharp ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 167: Line 198:
{{Mapping|legend=1| 1 1 2 1 | 0 2 1 6 }}
{{Mapping|legend=1| 1 1 2 1 | 0 2 1 6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 357.938
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1202.488{{c}}, ~5/4 = 358.680{{c}}
{{Multival|legend=1| 2 1 6 -3 4 11 }}
: [[error map]]: {{val| +2.488 +17.893 -22.658 -14.258 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 358.495{{c}}
: error map: {{val| 0.000 +15.035 -27.818 -17.854 }}


{{Optimal ET sequence|legend=1| 3d, 7d, 10, 37cd, 47bccd, 57bccdd }}
{{Optimal ET sequence|legend=1| 3d, 7d, 10 }}


[[Badness]]: 0.028942
[[Badness]] (Sintel): 0.732


=== 11-limit ===
=== 11-limit ===
Line 182: Line 215:
Mapping: {{mapping| 1 1 2 1 2 | 0 2 1 6 5 }}
Mapping: {{mapping| 1 1 2 1 2 | 0 2 1 6 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 356.106
Optimal tunings:
* WE: ~2 = 1201.518{{c}}, ~5/4 = 356.557{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 356.457{{c}}


{{Optimal ET sequence|legend=1| 3de, 7d, 10, 17d, 27cde }}
{{Optimal ET sequence|legend=0| 3de, 7d, 10, 17d }}


Badness: 0.022366
Badness (Sintel): 0.739


== Dichotic ==
== Dichotic ==
In dichotic, 7/4 is found at a stack of two perfect fourths.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 195: Line 232:
{{Mapping|legend=1| 1 1 2 4 | 0 2 1 -4 }}
{{Mapping|legend=1| 1 1 2 4 | 0 2 1 -4 }}


{{Multival|legend=1| 2 1 -4 -3 -12 -12 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.802{{c}}, ~5/4 = 356.502{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 356.264
: [[error map]]: {{val| +0.802 +11.851 -28.208 +8.374 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.275{{c}}
: error map: {{val| 0.000 +10.595 -30.039 +6.074 }}


{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c, 37c, 64bccc }}
{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c }}


[[Badness]]: 0.037565
[[Badness]] (Sintel): 0.951


=== 11-limit ===
=== 11-limit ===
Line 210: Line 249:
Mapping: {{mapping| 1 1 2 4 2 | 0 2 1 -4 5 }}
Mapping: {{mapping| 1 1 2 4 2 | 0 2 1 -4 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.262
Optimal tunings:
* WE: ~2 = 1199.504{{c}}, ~5/4 = 354.115{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.236{{c}}


{{Optimal ET sequence|legend=1| 7, 10, 17, 27ce, 44cce }}
{{Optimal ET sequence|legend=0| 7, 10, 17 }}


Badness: 0.030680
Badness (Sintel): 1.01


==== 13-limit ====
==== 13-limit ====
Line 223: Line 264:
Mapping: {{mapping| 1 1 2 4 2 4 | 0 2 1 -4 5 -1 }}
Mapping: {{mapping| 1 1 2 4 2 4 | 0 2 1 -4 5 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.365
Optimal tunings:
* WE: ~2 = 1199.289{{c}}, ~5/4 = 354.156{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.340{{c}}


{{Optimal ET sequence|legend=1| 7, 10, 17, 27ce, 44cce }}
{{Optimal ET sequence|legend=0| 7, 10, 17, 27ce, 44cce }}


Badness: 0.021674
Badness (Sintel): 0.896


=== Dichotomic ===
=== Dichotomic ===
Line 236: Line 279:
Mapping: {{mapping| 1 1 2 4 4 | 0 2 1 -4 -2 }}
Mapping: {{mapping| 1 1 2 4 4 | 0 2 1 -4 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.073
Optimal tunings:
* WE: ~2 = 1203.949{{c}}, ~5/4 = 355.239{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.024{{c}}


{{Optimal ET sequence|legend=1| 3, 7, 10e, 17e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e }}


Badness: 0.031719
Badness (Sintel): 1.05


==== 13-limit ====
==== 13-limit ====
Line 249: Line 294:
Mapping: {{mapping| 1 1 2 4 4 4 | 0 2 1 -4 -2 -1 }}
Mapping: {{mapping| 1 1 2 4 4 4 | 0 2 1 -4 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.313
Optimal tunings:
* WE: ~2 = 1202.979{{c}}, ~5/4 = 355.193{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.254{{c}}


{{Optimal ET sequence|legend=1| 3, 7, 10e, 17e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e }}


Badness: 0.022741
Badness (Sintel): 0.940


=== Dichosis ===
=== Dichosis ===
Line 262: Line 309:
Mapping: {{mapping| 1 1 2 4 5 | 0 2 1 -4 -5 }}
Mapping: {{mapping| 1 1 2 4 5 | 0 2 1 -4 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 360.659
Optimal tunings:
* WE: ~2 = 1197.526{{c}}, ~5/4 = 359.915{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.745{{c}}


{{Optimal ET sequence|legend=1| 3, 7e, 10 }}
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}


Badness: 0.041361
Badness (Sintel): 1.37


==== 13-limit ====
==== 13-limit ====
Line 275: Line 324:
Mapping: {{mapping| 1 1 2 4 5 4 | 0 2 1 -4 -5 -1 }}
Mapping: {{mapping| 1 1 2 4 5 4 | 0 2 1 -4 -5 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 360.646
Optimal tunings:
* WE: ~2 = 1197.922{{c}}, ~5/4 = 360.021{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.722{{c}}


{{Optimal ET sequence|legend=1| 3, 7e, 10 }}
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}


Badness: 0.027938
Badness (Sintel): 1.15


== Decimal ==
== Decimal ==
{{Main| Decimal }}
{{Main| Decimal }}
{{See also| Jubilismic clan }}
Decimal tempers out 49/48 and [[50/49]], and has a semi-octave period for 7/5~10/7 and a hemitwelfth generator for 7/4~12/7. Its ploidacot is diploid dicot. [[10edo]] makes for a good tuning, from which it derives its name. [[14edo]] in the 14c val and [[24edo]] in the 24c val are also among the possibilities.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 292: Line 346:
: mapping generators: ~7/5, ~7/4
: mapping generators: ~7/5, ~7/4


{{Multival|legend=1| 4 2 2 -6 -8 -1 }}
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 603.286{{c}}, ~7/4 = 953.637{{c}} (~7/6 = 252.935{{c}})
: [[error map]]: {{val| +6.571 +5.318 -22.821 -2.047 }}
* [[CWE]]: ~7/5 = 600.000{{c}}, ~7/4 = 950.957{{c}} (~7/6 = 249.043{{c}})
: error map: {{val| 0.000 -0.041 -35.357 -17.869 }}


[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~7/4 = 948.443 (~7/6 = 251.557)
{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd }}


{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd, 62cccdd }}
[[Badness]] (Sintel): 0.717
 
[[Badness]]: 0.028334


=== 11-limit ===
=== 11-limit ===
Line 307: Line 363:
Mapping: {{mapping| 2 0 3 4 -1 | 0 2 1 1 5 }}
Mapping: {{mapping| 2 0 3 4 -1 | 0 2 1 1 5 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 946.507 (~7/6 = 253.493)
Optimal tunings:
* WE: ~7/5 = 603.558{{c}}, ~7/4 = 952.121{{c}} (~7/6 = 254.996{{c}})
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 948.610{{c}} (~7/6 = 251.390{{c}})
 
{{Optimal ET sequence|legend=0| 4e, 10, 14c, 24c }}
 
Badness (Sintel): 0.883
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 25/24, 45/44, 49/48, 91/90


{{Optimal ET sequence|legend=1| 10, 14c, 24c, 38ccd, 52cccde }}
Mapping: {{mapping| 2 0 3 4 -1 1| 0 2 1 1 5 4}}


Badness: 0.026712
Optimal tunings:
* WE: ~7/5 = 603.612{{c}}, ~7/4 = 953.663{{c}} (~7/6 = 253.562{{c}})
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 950.116{{c}} (~7/6 = 249.884{{c}})
 
{{Optimal ET sequence|legend=0| 4ef, 10, 14cf, 24cf }}
 
Badness (Sintel): 0.881


=== Decimated ===
=== Decimated ===
Line 320: Line 393:
Mapping: {{mapping| 2 0 3 4 10 | 0 2 1 1 -2 }}
Mapping: {{mapping| 2 0 3 4 10 | 0 2 1 1 -2 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 944.934 (~7/6 = 255.066)
Optimal tunings:
* WE: ~7/5 = 604.535{{c}}, ~7/4 = 952.076{{c}} (~7/6 = 256.994{{c}})
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 946.108{{c}} (~7/6 = 253.892{{c}})


{{Optimal ET sequence|legend=1| 4, 10e, 14c }}
{{Optimal ET sequence|legend=0| 4, 10e, 14c }}


Badness: 0.031456
Badness (Sintel): 1.04


=== Decibel ===
=== Decibel ===
Line 333: Line 408:
Mapping: {{mapping| 2 0 3 4 7 | 0 2 1 1 0 }}
Mapping: {{mapping| 2 0 3 4 7 | 0 2 1 1 0 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 956.507 (~8/7 = 243.493)
Optimal tunings:
* WE: ~7/5 = 599.404{{c}}, ~7/4 = 955.557{{c}} (~8/7 = 243.251{{c}})
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 956.169{{c}} (~8/7 = 243.831{{c}})


{{Optimal ET sequence|legend=1| 4, 6, 10 }}
{{Optimal ET sequence|legend=0| 4, 6, 10 }}


Badness: 0.032385
Badness (Sintel): 1.07


== Sidi ==
== Sidi ==
Sidi tempers out 245/243, and splits 5/2~12/5 in two. Its ploidacot is beta-tetracot. This relates it to [[squares]], to which it can be used as a simpler alternative. 14edo in the 14c val can be used as a tuning, in which case it is identical to squares, however.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 25/24, 245/243
[[Comma list]]: 25/24, 245/243


{{Mapping|legend=1| 1 3 3 6 | 0 -4 -2 -9 }}
{{Mapping|legend=1| 1 -1 1 -3 | 0 4 2 9 }}
 
: mapping generators: ~2, ~9/7


{{Multival|legend=1| 4 2 9 -12 3 15 }}
: mapping generators: ~2, ~14/9


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 427.208
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1207.178{{c}}, ~14/9 = 777.414{{c}}
: [[error map]]: {{val| +7.178 +0.523 -24.308 +6.367 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~14/9 = 773.872{{c}}
: error map: {{val| 0.000 -6.464 -38.569 -3.973 }}


{{Optimal ET sequence|legend=1| 3d, 14c, 45cc, 59bcccd }}
{{Optimal ET sequence|legend=1| 3d, …, 11cd, 14c }}


[[Badness]]: 0.056586
[[Badness]] (Sintel): 1.43


=== 11-limit ===
=== 11-limit ===
Line 361: Line 442:
Comma list: 25/24, 45/44, 99/98
Comma list: 25/24, 45/44, 99/98


Mapping: {{mapping| 1 3 3 6 7 | 0 -4 -2 -9 -10 }}
Mapping: {{mapping| 1 -1 1 -3 -3 | 0 4 2 9 10 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 427.273
Optimal tunings:
* WE: ~2 = 1207.200{{c}}, ~11/7 = 777.363{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/7 = 773.777{{c}}


{{Optimal ET sequence|legend=1| 3de, 14c, 45cce, 59bcccdee }}
{{Optimal ET sequence|legend=0| 3de, …, 11cdee, 14c }}


Badness: 0.032957
Badness (Sintel): 1.09


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Dicot family| ]] <!-- main article -->
[[Category:Dicot family| ]] <!-- main article -->
[[Category:Dicot| ]] <!-- key article -->
[[Category:Dicot| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 14:10, 17 August 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The dicot family of temperaments tempers out 25/24, the classical chromatic semitone.

Dicot

The head of this family, dicot, is generated by a classical third (major and minor mean the same thing), and two such thirds give a fifth. In fact, (5/4)2 = (3/2)(25/24). Its ploidacot is the same as its name, dicot.

Possible tunings for dicot are 7edo, 10edo, 17edo, 24edo using the val ⟨24 38 55] (24c), and 31edo using the val ⟨31 49 71] (31c). In a sense, what dicot is all about is using neutral thirds and sixths and pretending that these are 5-limit, and like any temperament which seems to involve a lot of "pretending", dicot is close to the edge of what can be sensibly called a temperament at all. In other words, it is an exotemperament.

Subgroup: 2.3.5

Comma list: 25/24

Mapping: [⟨1 1 2], ⟨0 2 1]]

mapping generators: ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1206.283 ¢, ~6/5 = 350.420 ¢
error map: ⟨+6.283 +5.167 -23.328]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 351.086 ¢
error map: ⟨0.000 +0.216 -35.228]

Tuning ranges:

Optimal ET sequence: 3, 4, 7, 17, 24c, 31c

Badness (Sintel): 0.306

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. Septimal dicot adds 36/35, flattie adds 21/20, sharpie adds 28/27, and dichotic adds 64/63, all retaining the same period and generator.

Decimal adds 49/48, sidi adds 245/243, and jamesbond adds 16/15. Here decimal divides the period to a semi-octave, and sidi uses 14/9 as a generator, with two of them making up the combined 5/2~12/5 neutral tenth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.

Temperaments discussed elsewhere are:

The rest are considered below.

2.3.5.11 subgroup

The 2.3.5.11-subgroup extension maps 11/9~27/22 to the neutral third. As such, it is related to most of the septimal extensions.

Subgroup: 2.3.5.11

Comma list: 25/24, 45/44

Subgroup val mapping: [⟨1 1 2 2], ⟨0 2 1 5]]

Gencom mapping: [⟨1 1 2 0 2], ⟨0 2 1 0 5]]

Optimal tunings:

  • WE: ~2 = 1206.750 ¢, ~6/5 = 348.684 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 348.954 ¢

Optimal ET sequence: 3e, 4e, 7, 24c, 31c

Badness (Sintel): 0.370

2.3.5.11.13 subgroup

Subgroup: 2.3.5.11.13

Comma list: 25/24, 40/39, 45/44

Subgroup val mapping: [⟨1 1 2 2 4], ⟨0 2 1 5 -1]]

Gencom mapping: [⟨1 1 2 0 2 4], ⟨0 2 1 0 5 -1]]

Optimal tunings:

  • WE: ~2 = 1202.433 ¢, ~5/4 = 351.237 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 350.978 ¢

Optimal ET sequence: 3e, 7, 17

Badness (Sintel): 0.536

Septimal dicot

Septimal dicot is the extension where 7/6 and 9/7 are also conflated into 5/4~6/5. Although 5/4~6/5 covers a giant block of pitches already, 7/6 and 9/7 are often considered as thirds too. On that account one could argue for the canonicity of this extension, despite the relatively poor accuracy.

Subgroup: 2.3.5.7

Comma list: 15/14, 25/24

Mapping: [⟨1 1 2 2], ⟨0 2 1 3]]

Optimal tunings:

  • WE: ~2 = 1205.532 ¢, ~6/5 = 337.931 ¢
error map: ⟨+5.532 -20.561 -37.319 +56.032]
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 338.561 ¢
error map: ⟨0.000 -24.834 -47.753 +46.856]

Optimal ET sequence: 3d, 4, 7

Badness (Sintel): 0.504

11-limit

Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 25/24

Mapping: [⟨1 1 2 2 2], ⟨0 2 1 3 5]]

Optimal tunings:

  • WE: ~2 = 1203.346 ¢, ~6/5 = 343.078 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 343.260 ¢

Optimal ET sequence: 3de, 4e, 7

Badness (Sintel): 0.656

Eudicot

Subgroup: 2.3.5.7.11

Comma list: 15/14, 25/24, 33/32

Mapping: [⟨1 1 2 2 4], ⟨0 2 1 3 -2]]

Optimal tunings:

  • WE: ~2 = 1205.828 ¢, ~6/5 = 337.683 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 336.909 ¢

Optimal ET sequence: 3d, 4, 7, 18bc, 25bccd

Badness (Sintel): 0.896

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 15/14, 25/24, 33/32, 40/39

Mapping: [⟨1 1 2 2 4 4], ⟨0 2 1 3 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1202.660 ¢, ~6/5 = 339.597 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 339.104 ¢

Optimal ET sequence: 3d, 4, 7

Badness (Sintel): 0.985

Flattie

This temperament used to be known as flat. Unlike septimal dicot where 7/6 is added to the neutral third, here 8/7 is added instead.

Subgroup: 2.3.5.7

Comma list: 21/20, 25/24

Mapping: [⟨1 1 2 3], ⟨0 2 1 -1]]

Optimal tunings:

  • WE: ~2 = 1220.466 ¢, ~6/5 = 337.577 ¢
error map: ⟨+20.466 -6.335 -7.804 -45.004]
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 335.391 ¢
error map: ⟨0.000 -31.173 -50.922 -104.217]

Optimal ET sequence: 3, 4, 7d, 11cd, 18bcddd

Badness (Sintel): 0.642

11-limit

Subgroup: 2.3.5.7.11

Comma list: 21/20, 25/24, 33/32

Mapping: [⟨1 1 2 3 4], ⟨0 2 1 -1 -2]]

Optimal tunings:

  • WE: ~2 = 1216.069 ¢, ~6/5 = 342.052 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 338.467 ¢

Optimal ET sequence: 3, 4, 7d

Badness (Sintel): 0.826

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 14/13, 21/20, 25/24, 33/32

Mapping: [⟨1 1 2 3 4 4], ⟨0 2 1 -1 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1211.546 ¢, ~6/5 = 344.304 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 341.373 ¢

Optimal ET sequence: 3, 4, 7d

Badness (Sintel): 0.968

Sharpie

This temperament used to be known as sharp. This is where you find 7/6 at the major second and 7/4 at the major sixth.

Subgroup: 2.3.5.7

Comma list: 25/24, 28/27

Mapping: [⟨1 1 2 1], ⟨0 2 1 6]]

Optimal tunings:

  • WE: ~2 = 1202.488 ¢, ~5/4 = 358.680 ¢
error map: ⟨+2.488 +17.893 -22.658 -14.258]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 358.495 ¢
error map: ⟨0.000 +15.035 -27.818 -17.854]

Optimal ET sequence: 3d, 7d, 10

Badness (Sintel): 0.732

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 28/27, 35/33

Mapping: [⟨1 1 2 1 2], ⟨0 2 1 6 5]]

Optimal tunings:

  • WE: ~2 = 1201.518 ¢, ~5/4 = 356.557 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 356.457 ¢

Optimal ET sequence: 3de, 7d, 10, 17d

Badness (Sintel): 0.739

Dichotic

In dichotic, 7/4 is found at a stack of two perfect fourths.

Subgroup: 2.3.5.7

Comma list: 25/24, 64/63

Mapping: [⟨1 1 2 4], ⟨0 2 1 -4]]

Optimal tunings:

  • WE: ~2 = 1200.802 ¢, ~5/4 = 356.502 ¢
error map: ⟨+0.802 +11.851 -28.208 +8.374]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 356.275 ¢
error map: ⟨0.000 +10.595 -30.039 +6.074]

Optimal ET sequence: 3, 7, 10, 17, 27c

Badness (Sintel): 0.951

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 64/63

Mapping: [⟨1 1 2 4 2], ⟨0 2 1 -4 5]]

Optimal tunings:

  • WE: ~2 = 1199.504 ¢, ~5/4 = 354.115 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.236 ¢

Optimal ET sequence: 7, 10, 17

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 40/39, 45/44, 64/63

Mapping: [⟨1 1 2 4 2 4], ⟨0 2 1 -4 5 -1]]

Optimal tunings:

  • WE: ~2 = 1199.289 ¢, ~5/4 = 354.156 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.340 ¢

Optimal ET sequence: 7, 10, 17, 27ce, 44cce

Badness (Sintel): 0.896

Dichotomic

Subgroup: 2.3.5.7.11

Comma list: 22/21, 25/24, 33/32

Mapping: [⟨1 1 2 4 4], ⟨0 2 1 -4 -2]]

Optimal tunings:

  • WE: ~2 = 1203.949 ¢, ~5/4 = 355.239 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.024 ¢

Optimal ET sequence: 3, 7, 10e

Badness (Sintel): 1.05

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 22/21, 25/24, 33/32, 40/39

Mapping: [⟨1 1 2 4 4 4], ⟨0 2 1 -4 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1202.979 ¢, ~5/4 = 355.193 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.254 ¢

Optimal ET sequence: 3, 7, 10e

Badness (Sintel): 0.940

Dichosis

Subgroup: 2.3.5.7.11

Comma list: 25/24, 35/33, 64/63

Mapping: [⟨1 1 2 4 5], ⟨0 2 1 -4 -5]]

Optimal tunings:

  • WE: ~2 = 1197.526 ¢, ~5/4 = 359.915 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 360.745 ¢

Optimal ET sequence: 3, 7e, 10

Badness (Sintel): 1.37

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 35/33, 40/39, 64/63

Mapping: [⟨1 1 2 4 5 4], ⟨0 2 1 -4 -5 -1]]

Optimal tunings:

  • WE: ~2 = 1197.922 ¢, ~5/4 = 360.021 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 360.722 ¢

Optimal ET sequence: 3, 7e, 10

Badness (Sintel): 1.15

Decimal

Main article: Decimal
See also: Jubilismic clan

Decimal tempers out 49/48 and 50/49, and has a semi-octave period for 7/5~10/7 and a hemitwelfth generator for 7/4~12/7. Its ploidacot is diploid dicot. 10edo makes for a good tuning, from which it derives its name. 14edo in the 14c val and 24edo in the 24c val are also among the possibilities.

Subgroup: 2.3.5.7

Comma list: 25/24, 49/48

Mapping: [⟨2 0 3 4], ⟨0 2 1 1]]

mapping generators: ~7/5, ~7/4

Optimal tunings:

  • WE: ~7/5 = 603.286 ¢, ~7/4 = 953.637 ¢ (~7/6 = 252.935 ¢)
error map: ⟨+6.571 +5.318 -22.821 -2.047]
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 950.957 ¢ (~7/6 = 249.043 ¢)
error map: ⟨0.000 -0.041 -35.357 -17.869]

Optimal ET sequence: 4, 10, 14c, 24c, 38ccd

Badness (Sintel): 0.717

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 49/48

Mapping: [⟨2 0 3 4 -1], ⟨0 2 1 1 5]]

Optimal tunings:

  • WE: ~7/5 = 603.558 ¢, ~7/4 = 952.121 ¢ (~7/6 = 254.996 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 948.610 ¢ (~7/6 = 251.390 ¢)

Optimal ET sequence: 4e, 10, 14c, 24c

Badness (Sintel): 0.883

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 45/44, 49/48, 91/90

Mapping: [⟨2 0 3 4 -1 1], ⟨0 2 1 1 5 4]]

Optimal tunings:

  • WE: ~7/5 = 603.612 ¢, ~7/4 = 953.663 ¢ (~7/6 = 253.562 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 950.116 ¢ (~7/6 = 249.884 ¢)

Optimal ET sequence: 4ef, 10, 14cf, 24cf

Badness (Sintel): 0.881

Decimated

Subgroup: 2.3.5.7.11

Comma list: 25/24, 33/32, 49/48

Mapping: [⟨2 0 3 4 10], ⟨0 2 1 1 -2]]

Optimal tunings:

  • WE: ~7/5 = 604.535 ¢, ~7/4 = 952.076 ¢ (~7/6 = 256.994 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 946.108 ¢ (~7/6 = 253.892 ¢)

Optimal ET sequence: 4, 10e, 14c

Badness (Sintel): 1.04

Decibel

Subgroup: 2.3.5.7.11

Comma list: 25/24, 35/33, 49/48

Mapping: [⟨2 0 3 4 7], ⟨0 2 1 1 0]]

Optimal tunings:

  • WE: ~7/5 = 599.404 ¢, ~7/4 = 955.557 ¢ (~8/7 = 243.251 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 956.169 ¢ (~8/7 = 243.831 ¢)

Optimal ET sequence: 4, 6, 10

Badness (Sintel): 1.07

Sidi

Sidi tempers out 245/243, and splits 5/2~12/5 in two. Its ploidacot is beta-tetracot. This relates it to squares, to which it can be used as a simpler alternative. 14edo in the 14c val can be used as a tuning, in which case it is identical to squares, however.

Subgroup: 2.3.5.7

Comma list: 25/24, 245/243

Mapping: [⟨1 -1 1 -3], ⟨0 4 2 9]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1207.178 ¢, ~14/9 = 777.414 ¢
error map: ⟨+7.178 +0.523 -24.308 +6.367]
  • CWE: ~2 = 1200.000 ¢, ~14/9 = 773.872 ¢
error map: ⟨0.000 -6.464 -38.569 -3.973]

Optimal ET sequence: 3d, …, 11cd, 14c

Badness (Sintel): 1.43

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 99/98

Mapping: [⟨1 -1 1 -3 -3], ⟨0 4 2 9 10]]

Optimal tunings:

  • WE: ~2 = 1207.200 ¢, ~11/7 = 777.363 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/7 = 773.777 ¢

Optimal ET sequence: 3de, …, 11cdee, 14c

Badness (Sintel): 1.09

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